Order = 244823040 = 210.33.5.7.11.23.
Mult = 1.
Out = 1.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of M24 are a, b where a is in class 2B, b is in class 3A, ab has order 23 and abababbababbabb has order 4.

## Black box algorithms

### Finding generators

Group Algorithm File

### Checking generators (semi-presentations)

Group Semi-presentation File
M24 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 23, o(abababbababbabb) = 4, o(abababb) = 12 〉〉 Download

## Presentations

M24 a, b | a2 = b3 = (ab)23 = [a, b]12 = [a, bab]5 = (ababab−1)3(abab−1ab−1)3 = (ab(abab−1)3)4 = 1 〉 Details

## Maximal subgroups

### Maximal subgroups of M24

Subgroup Order Index Programs/reps
M23 10 200 960 24 Program: Standard generators
Program: Generators
M22:2 887 040 276 Program: Generators
24:A8 322 560 759 Program: Generators
M12:2 190 080 1 288 Program: Generators
26:3.S6 138 240 1 771 Program: Generators
L3(4):S3 120 960 2 024 Program: Generators
26:(L3(2) × S3) 64 512 3 795 Program: Generators
L2(23) 6 072 40 320 Program: Generators
L2(7) 168 1 457 280 Program: Generators

## Conjugacy classes

### Notes

(M24) Problems of algebraic conjugacy are not yet dealt with.

### Conjugacy classes of M24

Conjugacy class Centraliser order Power up Class rep(s)
1A 244 823 040 Omitted owing to length.
2A 21 504 4A 4B 6A 8A 12A 14A 14B ababababbababababbababababbababababb
2B 7 680 4C 6B 10A 12B (ababbababb)3
3A 1 080 6A 12A 15A 15B abababbabababbabababbabababb
3B 504 6B 12B 21A 21B ababbababbababbababb
4A 384 12A (abababb)3
4B 128 8A ababababbababababb
4C 96 12B (ababb)3
5A 60 10A 15A 15B abababbababbabababbababb
6A 24 12A abababbabababb
6B 24 12B ababbababb
7A 42 7B3 14A 14B 21A 21B (ababababbabbababababbabb)3
7B 42 7A3 14A 14B 21A 21B ababababbabbababababbabb
8A 16 ababababb
10A 20 abababbababb
11A 11 ababababbabbabb
12A 12 abababb
12B 12 ababb
14A 14 14B3 (ababababbabb)3
14B 14 14A3 ababababbabb
15A 15 15B7 abababababb
15B 15 15A7 (abababababb)7
21A 21 21B5 ababababababb
21B 21 21A5 (ababababababb)5
23A 23 23B5 ab
23B 23 23A5 (ab)5